mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.CofilteredSystem

Goal state before:
pp: J : Type u
inst✝ : Category.{v_1, u} J
F : J ⥤ Type v
i j : J
h : F.IsMittagLeffler
f : j ⟶ i
k : J
g : k ⟶ j
hg : F.eventualRange j = range ⇑(ConcreteCategory.hom (F.map g))
⊢ F.eventualRange i ⊆ ⇑(ConcreteCategory.hom (F.map f)) '' range ⇑(ConcreteCategory.hom (F.map g))

Tactic:
intro x

Goal state after:
J : Type u
inst✝ : Category.{v_1, u} J
F : J ⥤ Type v
i j : J
h : F.IsMittagLeffler
f : j ⟶ i
k : J
g : k ⟶ j
hg : F.eventualRange j = range ⇑(ConcreteCategory.hom (F.map g))
x : F.obj i
⊢ x ∈ F.eventualRange i → x ∈ ⇑(ConcreteCategory.hom (F.map f)) '' range ⇑(ConcreteCategory.hom (F.map g))
Module: Mathlib.Analysis.Convex.Approximation

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
φ : E → ℝ
inst✝⁹ : RCLike 𝕜
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : IsScalarTower ℝ 𝕜 E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : HereditarilyLindelofSpace E
hφc : LowerSemicontinuous φ
hφcv : ConvexOn ℝ univ φ
⊢ ∃ l c, (∀ (i : ℕ), ⇑re ∘ ⇑(l i) + const E (c i) ≤ φ) ∧ ⨆ i, ⇑re ∘ ⇑(l i) + const E (c i) = φ

Tactic:
obtain ⟨l, c, hle, hsup⟩ := hφcv.sSup_of_nat_affine_eq (𝕜 := 𝕜) (s := univ) isClosed_univ
    (lowerSemicontinuousOn_univ_iff.2 hφc)
  refine ⟨l, c, fun i x ↦ hle i ⟨x, trivial⟩, ?_⟩
  ext x
  simpa using congrFun hsup ⟨x, trivial⟩

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Stochastic

Goal state before:
pp: R : Type u_1
n : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : Semiring R
inst✝¹ : PartialOrder R
inst✝ : IsOrderedRing R
⊢ Convex R ↑(rowStochastic R n)

Tactic:
intro x

Goal state after:
R : Type u_1
n : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : Semiring R
inst✝¹ : PartialOrder R
inst✝ : IsOrderedRing R
x : Matrix n n R
⊢ x ∈ ↑(rowStochastic R n) → StarConvex R x ↑(rowStochastic R n)
Module: Mathlib.LinearAlgebra.Matrix.Stochastic

Goal state before:
pp: R : Type u_1
n : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : Semiring R
inst✝¹ : PartialOrder R
inst✝ : IsOrderedRing R
⊢ Convex R ↑(colStochastic R n)

Tactic:
intro x

Goal state after:
R : Type u_1
n : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : Semiring R
inst✝¹ : PartialOrder R
inst✝ : IsOrderedRing R
x : Matrix n n R
⊢ x ∈ ↑(colStochastic R n) → StarConvex R x ↑(colStochastic R n)
Module: Mathlib.Analysis.Convex.DoublyStochasticMatrix

Goal state before:
pp: R : Type u_1
n : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : Semiring R
inst✝¹ : PartialOrder R
inst✝ : IsOrderedRing R
⊢ Convex R ↑(doublyStochastic R n)

Tactic:
intro x

Goal state after:
R : Type u_1
n : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : Semiring R
inst✝¹ : PartialOrder R
inst✝ : IsOrderedRing R
x : Matrix n n R
⊢ x ∈ ↑(doublyStochastic R n) → StarConvex R x ↑(doublyStochastic R n)
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Ring R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : IsTorsionFree R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
r₁ : R
hr₁0 : 0 < r₁
hr₁1 : r₁ < 1
h₁ : Sbtw R (t.points i₂) ((lineMap (t.points i₂) (t.points i₃)) r₁) (t.points i₃)
r₂ : R
hr₂0 : 0 < r₂
hr₂1 : r₂ < 1
h₂ : Sbtw R (t.points i₁) ((lineMap (t.points i₁) (t.points i₃)) r₂) (t.points i₃)
w : Fin (2 + 1) → R
hw : ∑ i, w i = 1
hp : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R (Set.range t.points)
h₁' :
  (Finset.affineCombination R Finset.univ t.points) w ∈
    line[R, (Finset.affineCombination R Finset.univ t.points) (Pi.single i₁ 1),
      (Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i₂ i₃ r₁)]
h₂' :
  (Finset.affineCombination R Finset.univ t.points) w ∈ line[R, t.points i₂, (lineMap (t.points i₁) (t.points i₃)) r₂]
h₁s : SignType.sign (w i₂) = SignType.sign (w i₃)
h₂s : SignType.sign (w i₁) = SignType.sign (w i₃)
hs : ∀ (i : Fin 3), SignType.sign (w i) = SignType.sign (w i₃)
hss : SignType.sign (∑ i, w i) = 1
hs' : SignType.sign (∑ i, w i) = SignType.sign (w i₃)
⊢ w i₁ ∈ Set.Ioo 0 1

Tactic:
hs'

Goal state after:
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Ring R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : IsTorsionFree R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
r₁ : R
hr₁0 : 0 < r₁
hr₁1 : r₁ < 1
h₁ : Sbtw R (t.points i₂) ((lineMap (t.points i₂) (t.points i₃)) r₁) (t.points i₃)
r₂ : R
hr₂0 : 0 < r₂
hr₂1 : r₂ < 1
h₂ : Sbtw R (t.points i₁) ((lineMap (t.points i₁) (t.points i₃)) r₂) (t.points i₃)
w : Fin (2 + 1) → R
hw : ∑ i, w i = 1
hp : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R (Set.range t.points)
h₁' :
  (Finset.affineCombination R Finset.univ t.points) w ∈
    line[R, (Finset.affineCombination R Finset.univ t.points) (Pi.single i₁ 1),
      (Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i₂ i₃ r₁)]
h₂' :
  (Finset.affineCombination R Finset.univ t.points) w ∈ line[R, t.points i₂, (lineMap (t.points i₁) (t.points i₃)) r₂]
h₁s : SignType.sign (w i₂) = SignType.sign (w i₃)
h₂s : SignType.sign (w i₁) = SignType.sign (w i₃)
hs : ∀ (i : Fin 3), SignType.sign (w i) = SignType.sign (w i₃)
hss : SignType.sign (w i₃) = 1
hs' : SignType.sign (∑ i, w i) = SignType.sign (w i₃)
⊢ w i₁ ∈ Set.Ioo 0 1
Module: Mathlib.Analysis.Convex.Birkhoff

Goal state before:
pp: R : Type u_1
n : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : Field R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
h✝ : Nonempty n
d : ℕ
ih :
  ∀ m < d,
    ∀ (M : Matrix n n R) (s : R),
      0 ≤ s →
        (∃ M' ∈ doublyStochastic R n, M = s • M') →
          #{i | M i.1 i.2 ≠ 0} = m → ∃ w, (∀ (σ : Equiv.Perm n), 0 ≤ w σ) ∧ ∑ σ, w σ • Equiv.Perm.permMatrix R σ = M
M : Matrix n n R
s : R
hs : 0 ≤ s
hM : (∀ (i j : n), 0 ≤ M i j) ∧ (∀ (i : n), ∑ j, M i j = s) ∧ ∀ (j : n), ∑ i, M i j = s
hd : #{i | M i.1 i.2 ≠ 0} = d
hs'✝ : 0 < s
σ : Equiv.Perm n
hσ : ∀ (i j : n), M i j = 0 → Equiv.Perm.permMatrix R σ i j = 0
i : n
hi : i ∈ univ
hi' : ∀ x' ∈ univ, M i (σ i) ≤ M x' (σ x')
N : Matrix n n R := M - M i (σ i) • Equiv.Perm.permMatrix R σ
hMi' : 0 < M i (σ i)
s' : R := s - M i (σ i)
hs' : 0 ≤ s'
⊢ (∀ (i j : n), 0 ≤ N i j) ∧ (∀ (i : n), ∑ j, N i j = s') ∧ ∀ (j : n), ∑ i, N i j = s'

Tactic:
simp only [Matrix.sub_apply, Matrix.smul_apply, PEquiv.toMatrix_apply, Equiv.toPEquiv_apply,
      Option.mem_def, Option.some.injEq, smul_eq_mul, mul_ite, mul_one, mul_zero, sub_nonneg,
      sum_sub_distrib, sum_ite_eq, mem_univ, ↓reduceIte, N]

Goal state after:
R : Type u_1
n : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : Field R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
h✝ : Nonempty n
d : ℕ
ih :
  ∀ m < d,
    ∀ (M : Matrix n n R) (s : R),
      0 ≤ s →
        (∃ M' ∈ doublyStochastic R n, M = s • M') →
          #{i | M i.1 i.2 ≠ 0} = m → ∃ w, (∀ (σ : Equiv.Perm n), 0 ≤ w σ) ∧ ∑ σ, w σ • Equiv.Perm.permMatrix R σ = M
M : Matrix n n R
s : R
hs : 0 ≤ s
hM : (∀ (i j : n), 0 ≤ M i j) ∧ (∀ (i : n), ∑ j, M i j = s) ∧ ∀ (j : n), ∑ i, M i j = s
hd : #{i | M i.1 i.2 ≠ 0} = d
hs'✝ : 0 < s
σ : Equiv.Perm n
hσ : ∀ (i j : n), M i j = 0 → Equiv.Perm.permMatrix R σ i j = 0
i : n
hi : i ∈ univ
hi' : ∀ x' ∈ univ, M i (σ i) ≤ M x' (σ x')
N : Matrix n n R := M - M i (σ i) • Equiv.Perm.permMatrix R σ
hMi' : 0 < M i (σ i)
s' : R := s - M i (σ i)
hs' : 0 ≤ s'
⊢ (∀ (i_1 j : n), (if σ i_1 = j then M i (σ i) else 0) ≤ M i_1 j) ∧
    (∀ (i_1 : n), ∑ x, M i_1 x - M i (σ i) = s') ∧ ∀ (j : n), (∑ x, M x j - ∑ x, if σ x = j then M i (σ i) else 0) = s'
Module: Mathlib.Analysis.Convex.Body

Goal state before:
pp: case inr
V : Type u_1
inst✝³ : TopologicalSpace V
inst✝² : AddCommGroup V
inst✝¹ : Module ℝ V
inst✝ : ContinuousSMul ℝ V
K : ConvexBody V
h_zero : 0 ∈ K
a b : ℝ≥0
h : a ≤ b
ha : 0 < a
⊢ a • ↑K ⊆ b • ↑K

Tactic:
intro x

Goal state after:
case inr
V : Type u_1
inst✝³ : TopologicalSpace V
inst✝² : AddCommGroup V
inst✝¹ : Module ℝ V
inst✝ : ContinuousSMul ℝ V
K : ConvexBody V
h_zero : 0 ∈ K
a b : ℝ≥0
h : a ≤ b
ha : 0 < a
x : V
⊢ x ∈ a • ↑K → x ∈ b • ↑K
Module: Mathlib.Analysis.Convex.Body

Goal state before:
pp: case refine_1
V : Type u_1
inst✝² : SeminormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : T2Space V
u : ℕ → ℝ≥0
K : ConvexBody V
h_zero : 0 ∈ K
hu : ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → ‖↑(u n) - ↑0‖ < ε
x : V
h : x ∈ ⋂ n, (1 + ↑(u n)) • ↑K
C : ℝ
hC_pos : C > 0
hC_bdd : ∀ x ∈ ↑K, ‖x‖ ≤ C
⊢ ∀ (ε : ℝ), 0 < ε → ∃ b ∈ ↑K, ‖x - b‖ < ε

Tactic:
intro ε hε

Goal state after:
case refine_1
V : Type u_1
inst✝² : SeminormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : T2Space V
u : ℕ → ℝ≥0
K : ConvexBody V
h_zero : 0 ∈ K
hu : ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → ‖↑(u n) - ↑0‖ < ε
x : V
h : x ∈ ⋂ n, (1 + ↑(u n)) • ↑K
C : ℝ
hC_pos : C > 0
hC_bdd : ∀ x ∈ ↑K, ‖x‖ ≤ C
ε : ℝ
hε : 0 < ε
⊢ ∃ b ∈ ↑K, ‖x - b‖ < ε
Module: Mathlib.Analysis.Convex.Caratheodory

Goal state before:
pp: case h₁
𝕜 : Type u_1
E : Type u
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
⊢ (convexHull 𝕜) s ⊆ ⋃ t, ⋃ (_ : ↑t ⊆ s), ⋃ (_ : AffineIndependent 𝕜 Subtype.val), (convexHull 𝕜) ↑t

Tactic:
intro x

Goal state after:
case h₁
𝕜 : Type u_1
E : Type u
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x : E
⊢ x ∈ (convexHull 𝕜) s → x ∈ ⋃ t, ⋃ (_ : ↑t ⊆ s), ⋃ (_ : AffineIndependent 𝕜 Subtype.val), (convexHull 𝕜) ↑t
Module: Mathlib.Analysis.Convex.Birkhoff

Goal state before:
pp: R : Type u_1
n : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : Field R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
h✝ : Nonempty n
d : ℕ
ih :
  ∀ m < d,
    ∀ (M : Matrix n n R) (s : R),
      0 ≤ s →
        (∃ M' ∈ doublyStochastic R n, M = s • M') →
          #{i | M i.1 i.2 ≠ 0} = m → ∃ w, (∀ (σ : Equiv.Perm n), 0 ≤ w σ) ∧ ∑ σ, w σ • Equiv.Perm.permMatrix R σ = M
M : Matrix n n R
s : R
hs : 0 ≤ s
hM : (∀ (i j : n), 0 ≤ M i j) ∧ (∀ (i : n), ∑ j, M i j = s) ∧ ∀ (j : n), ∑ i, M i j = s
hd : #{i | M i.1 i.2 ≠ 0} = d
hs'✝ : 0 < s
σ : Equiv.Perm n
hσ : ∀ (i j : n), M i j = 0 → Equiv.Perm.permMatrix R σ i j = 0
i : n
hi : i ∈ univ
hi' : ∀ x' ∈ univ, M i (σ i) ≤ M x' (σ x')
N : Matrix n n R := M - M i (σ i) • Equiv.Perm.permMatrix R σ
hMi' : 0 < M i (σ i)
s' : R := s - M i (σ i)
hs' : 0 ≤ s'
this : ∃ M' ∈ doublyStochastic R n, N = s' • M'
⊢ ∀ a ∈ univ, N a.1 a.2 ≠ 0 → M a.1 a.2 ≠ 0

Tactic:
· rintro ⟨i', j'⟩ _ hN' hM'
      have hσ' : σ i' ≠ j' := by
        simpa [Equiv.toPEquiv_apply] using hσ i' j' hM'
      exact hN' <| by simp [N, hM', hσ', Equiv.toPEquiv_apply]

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Dual

Goal state before:
pp: case mem
R : Type u_1
inst✝⁶ : CommSemiring R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsOrderedRing R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
p : M →ₗ[R] N →ₗ[R] R
s : Set M
x : N
hx : x ∈ dual p s
y _y : M
h : _y ∈ s
⊢ 0 ≤ (p _y) x

Tactic:
| mem _y h =>

Goal state after:
Module: Mathlib.Analysis.Convex.Cone.TensorProduct

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module ℝ E
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module ℝ F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
inst✝³ : T2Space F
inst✝² : FiniteDimensional ℝ F
inst✝¹ : ContinuousSMul ℝ F
inst✝ : LocallyConvexSpace ℝ F
C₁ : PointedCone ℝ E
C₂ : ProperCone ℝ F
h₁_gen : Submodule.span ℝ ↑C₁ = ⊤
s : Set E
hs_fin : s.Finite
hs_lin : LinearIndepOn ℝ id s
hs_span : hull ℝ s = C₁
this : Fintype ↑s
⊢ ↑(hull ℝ s) ⊆ ↑(Submodule.span ℝ s)

Tactic:
intro x

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module ℝ E
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module ℝ F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
inst✝³ : T2Space F
inst✝² : FiniteDimensional ℝ F
inst✝¹ : ContinuousSMul ℝ F
inst✝ : LocallyConvexSpace ℝ F
C₁ : PointedCone ℝ E
C₂ : ProperCone ℝ F
h₁_gen : Submodule.span ℝ ↑C₁ = ⊤
s : Set E
hs_fin : s.Finite
hs_lin : LinearIndepOn ℝ id s
hs_span : hull ℝ s = C₁
this : Fintype ↑s
x : E
⊢ x ∈ ↑(hull ℝ s) → x ∈ ↑(Submodule.span ℝ s)
Module: Mathlib.Analysis.Convex.Cone.TensorProduct

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module ℝ E
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module ℝ F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
inst✝³ : T2Space F
inst✝² : FiniteDimensional ℝ F
inst✝¹ : ContinuousSMul ℝ F
inst✝ : LocallyConvexSpace ℝ F
C₁ : PointedCone ℝ E
C₂ : ProperCone ℝ F
h₁_simp : C₁.IsSimplicial
h₁_gen : Submodule.span ℝ ↑C₁ = ⊤
⊢ C₁.minTensorProduct ↑C₂ = C₁.maxTensorProduct ↑C₂

Tactic:
by
  classical
  obtain ⟨s, hs_fin, hs_lin, hs_span⟩ := h₁_simp
  have : Fintype s := hs_fin.fintype
  -- The conic hull (R≥0-span) is contained in the linear span (ℝ-span)
  have hull_sub_span : (hull ℝ s : Set E) ⊆ Submodule.span ℝ s := by
    intro x hx
    rw [SetLike.mem_coe, PointedCone.mem_hull_set] at hx
    obtain ⟨c, hc_supp, _, hc_sum⟩ := hx
    exact hc_sum ▸ Submodule.sum_mem _ fun m hm =>
      Submodule.smul_mem _ _ (Submodule.subset_span (hc_supp hm))
  -- Extract basis from `C₁.IsSimplicial` + generating
  let b := Basis.mk hs_lin <| by
    simpa only [id_eq, Subtype.range_coe] using!
      h₁_gen ▸ hs_span ▸ Submodule.span_le.mpr hull_sub_span
  -- Dual basis elements are in C₁*
  have h_coord_dual : ∀ i, b.coord i ∈ dual (Dual.eval ℝ E) C₁ :=
    basis_coord_mem_dual _ _ (hs_span ▸ (Submodule.span_mono <| by simp [b]))
  -- Reduce to proving z ∈ max → z ∈ min
  apply le_antisymm (minTensorProduct_le_maxTensorProduct C₁ C₂.toPointedCone)
  intro z hz
  -- Express z using basis: z = ∑ b_i ⊗ y_i
  rw [← (equivFinsuppOfBasisLeft b).symm_apply_apply z,
    TensorProduct.equivFinsuppOfBasisLeft_symm_apply, Finsupp.sum_fintype _ _ (by simp)]
  -- Show z ∈ min by showing b_i ∈ C₁ and y_i ∈ C₂
  refine Submodule.sum_mem _ fun i _ => tmul_mem_minTensorProduct ?_ ?_
  · simpa only [b, Basis.coe_mk] using! (hs_span ▸ subset_hull) i.prop
  · simp only [equivFinsuppOfBasisLeft_apply]
    rw [← ProperCone.dual_dual_flip (topDualPairing ℝ F) C₂]
    intro f (hf : (f : F →ₗ[ℝ] ℝ) ∈ dual (Dual.eval ℝ F) (C₂ : Set F))
    simp only [mem_maxTensorProduct] at hz
    have h_nonneg := hz (b.coord i) (h_coord_dual i) (f : F →ₗ[ℝ] ℝ) hf
    have h_eq : dualDistrib ℝ E F ((b.coord i) ⊗ₜ[ℝ] (f : F →ₗ[ℝ] ℝ)) =
        (f : F →ₗ[ℝ] ℝ) ∘ₗ (TensorProduct.lid ℝ F) ∘ₗ (b.coord i).rTensor F := by
      ext; simp [mul_comm]
    simpa only [h_eq, LinearMap.comp_apply, LinearEquiv.coe_coe] using! h_nonneg

Goal state after:
Module: Mathlib.Analysis.Convex.Extrema

Goal state before:
pp: E : Type u_1
β : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : TopologicalSpace E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : ContinuousSMul ℝ E
inst✝⁵ : AddCommGroup β
inst✝⁴ : PartialOrder β
inst✝³ : IsOrderedAddMonoid β
inst✝² : Module ℝ β
inst✝¹ : IsOrderedModule ℝ β
inst✝ : PosSMulReflectLE ℝ β
s : Set E
f : E → β
a : E
a_in_s : a ∈ s
h_localmin : IsLocalMinOn f s a
h_conv : ConvexOn ℝ s f
⊢ IsMinOn f s a

Tactic:
intro x

Goal state after:
E : Type u_1
β : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : TopologicalSpace E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : ContinuousSMul ℝ E
inst✝⁵ : AddCommGroup β
inst✝⁴ : PartialOrder β
inst✝³ : IsOrderedAddMonoid β
inst✝² : Module ℝ β
inst✝¹ : IsOrderedModule ℝ β
inst✝ : PosSMulReflectLE ℝ β
s : Set E
f : E → β
a : E
a_in_s : a ∈ s
h_localmin : IsLocalMinOn f s a
h_conv : ConvexOn ℝ s f
x : E
⊢ x ∈ s → x ∈ {x | (fun x ↦ f a ≤ f x) x}
Module: Mathlib.Analysis.Convex.Join

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t u v : Set E
⊢ convexJoin 𝕜 (convexJoin 𝕜 (convexJoin 𝕜 s t) u) v = convexJoin 𝕜 (convexJoin 𝕜 (convexJoin 𝕜 s u) t) v

Tactic:
convexJoin_right_comm

Goal state after:
Module: Mathlib.Analysis.Convex.Integral

Goal state before:
pp: case refine_3
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
g : α → E
hfi : Integrable g μ
hgm : StronglyMeasurable g
hfg : f =ᵐ[μ] g
this✝ : SeparableSpace ↑(range g ∩ s)
y₀ : E
h₀ : y₀ ∈ range g ∩ s
hg : ∀ᵐ (x : α) ∂μ, g x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (range g ∩ s)
G : ℕ → SimpleFunc α E := SimpleFunc.approxOn g ⋯ (range g ∩ s) y₀ h₀
this : Tendsto (fun n ↦ SimpleFunc.integral μ (G n)) atTop (𝓝 (∫ (x : α), g x ∂μ))
n : ℕ
⊢ ∀ (i : α), ↑(ContinuousLinearMap.id ℝ E) ((G n) i) ∈ s

Tactic:
intro x

Goal state after:
case refine_3
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
f : α → E
inst✝ : IsProbabilityMeasure μ
hs : Convex ℝ s
hsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hf : ∀ᵐ (x : α) ∂μ, f x ∈ s
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
g : α → E
hfi : Integrable g μ
hgm : StronglyMeasurable g
hfg : f =ᵐ[μ] g
this✝ : SeparableSpace ↑(range g ∩ s)
y₀ : E
h₀ : y₀ ∈ range g ∩ s
hg : ∀ᵐ (x : α) ∂μ, g x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (range g ∩ s)
G : ℕ → SimpleFunc α E := SimpleFunc.approxOn g ⋯ (range g ∩ s) y₀ h₀
this : Tendsto (fun n ↦ SimpleFunc.integral μ (G n)) atTop (𝓝 (∫ (x : α), g x ∂μ))
n : ℕ
x : α
⊢ ↑(ContinuousLinearMap.id ℝ E) ((G n) x) ∈ s
Module: Mathlib.Analysis.Convex.Intrinsic

Goal state before:
pp: case inr
𝕜 : Type u_1
V : Type u_2
W : Type u_3
Q : Type u_4
P : Type u_5
inst✝⁸ : Ring 𝕜
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module 𝕜 V
inst✝⁵ : TopologicalSpace P
inst✝⁴ : AddTorsor V P
inst✝³ : AddCommGroup W
inst✝² : Module 𝕜 W
inst✝¹ : TopologicalSpace Q
inst✝ : AddTorsor W Q
f : P → Q
s : Set P
e : [Nonempty ↑s] → ↥(affineSpan 𝕜 s) → ↥(affineSpan 𝕜 (f '' s))
he_homeo : ∀ [inst : Nonempty ↑s], IsHomeomorph e
he : ∀ [inst : Nonempty ↑s] (x : ↥(affineSpan 𝕜 s)), ↑(e x) = f ↑x
hs : s.Nonempty
⊢ intrinsicInterior 𝕜 (f '' s) = f '' intrinsicInterior 𝕜 s

Tactic:
have : Nonempty s := hs.to_subtype
    rw [intrinsicInterior, ← image_interior_preimage_comp e he_homeo,
      (funext he : (↑) ∘ e = f ∘ (↑)),
      preimage_image_eq_of_homeomorph_affineSpan e he_homeo he, image_comp]; rfl

Goal state after:
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: f : ℝ → ℝ
y : ℝ
hf_cvx : ConvexOn ℝ (Ici y) f
hf_cont : ContinuousWithinAt f (Ici y) y
⊢ ContinuousOn f (Ici y)

Tactic:
intro x

Goal state after:
f : ℝ → ℝ
y : ℝ
hf_cvx : ConvexOn ℝ (Ici y) f
hf_cont : ContinuousWithinAt f (Ici y) y
x : ℝ
⊢ x ∈ Ici y → ContinuousWithinAt f (Ici y) x
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: f : ℝ → ℝ
y : ℝ
hf_cvx : ConvexOn ℝ (Iic y) f
hf_cont : ContinuousWithinAt f (Iic y) y
⊢ ContinuousOn f (Iic y)

Tactic:
intro x

Goal state after:
f : ℝ → ℝ
y : ℝ
hf_cvx : ConvexOn ℝ (Iic y) f
hf_cont : ContinuousWithinAt f (Iic y) y
x : ℝ
⊢ x ∈ Iic y → ContinuousWithinAt f (Iic y) x
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: f : ℝ → ℝ
y z : ℝ
hf_cvx : ConvexOn ℝ (Ioc y z) f
hf_cont : ContinuousWithinAt f (Iic z) z
⊢ ContinuousOn f (Ioc y z)

Tactic:
intro x

Goal state after:
f : ℝ → ℝ
y z : ℝ
hf_cvx : ConvexOn ℝ (Ioc y z) f
hf_cont : ContinuousWithinAt f (Iic z) z
x : ℝ
⊢ x ∈ Ioc y z → ContinuousWithinAt f (Ioc y z) x
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: f : ℝ → ℝ
y z : ℝ
hf_cvx : ConvexOn ℝ (Ico y z) f
hf_cont : ContinuousWithinAt f (Ici y) y
⊢ ContinuousOn f (Ico y z)

Tactic:
intro x

Goal state after:
f : ℝ → ℝ
y z : ℝ
hf_cvx : ConvexOn ℝ (Ico y z) f
hf_cont : ContinuousWithinAt f (Ici y) y
x : ℝ
⊢ x ∈ Ico y z → ContinuousWithinAt f (Ico y z) x
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: f : ℝ → ℝ
y z : ℝ
hf_cvx : ConvexOn ℝ (Icc y z) f
hyz : y < z
hfy : ContinuousWithinAt f (Ici y) y
hfz : ContinuousWithinAt f (Iic z) z
this : ContinuousOn f (Ico y z) ∧ ContinuousOn f (Ioc y z)
⊢ ContinuousOn f (Icc y z)

Tactic:
intro x

Goal state after:
f : ℝ → ℝ
y z : ℝ
hf_cvx : ConvexOn ℝ (Icc y z) f
hyz : y < z
hfy : ContinuousWithinAt f (Ici y) y
hfz : ContinuousWithinAt f (Iic z) z
this : ContinuousOn f (Ico y z) ∧ ContinuousOn f (Ioc y z)
x : ℝ
⊢ x ∈ Icc y z → ContinuousWithinAt f (Icc y z) x
Module: Mathlib.Geometry.Convex.ConvexSpace.Defs

Goal state before:
pp: R✝ : Type u_1
X : Type u_2
M✝¹ : Type u_3
N✝¹ : Type u_4
P✝ : Type u_5
I : Type u_6
J : Type u_7
K : Type u_8
R : Type u
inst✝² : PartialOrder R
inst✝¹ : Semiring R
M✝ : Type u_9
N✝ : Type u_10
P : Type u_11
w : StdSimplex R M✝
x : M✝
inst✝ : IsStrictOrderedRing R
M : Type v
N : Type w
g : M → N
f : StdSimplex R M
⊢ ((mapDomain g f.weights).sum fun x r ↦ r) = 1

Tactic:
by simp [sum_mapDomain_index]

Goal state after:
Module: Mathlib.Geometry.Convex.ConvexSpace.Defs

Goal state before:
pp: R : Type u_1
M : Type u_3
inst✝³ : PartialOrder R
inst✝² : Semiring R
inst✝¹ : IsStrictOrderedRing R
inst✝ : ConvexSpace R M
s t : R
hs : 0 ≤ s
ht : 0 ≤ t
h : s + t = 1
x : M
⊢ convexCombPair s t hs ht h x x = x

Tactic:
unfold convexCombPair

Goal state after:
R : Type u_1
M : Type u_3
inst✝³ : PartialOrder R
inst✝² : Semiring R
inst✝¹ : IsStrictOrderedRing R
inst✝ : ConvexSpace R M
s t : R
hs : 0 ≤ s
ht : 0 ≤ t
h : s + t = 1
x : M
⊢ sConvexComb (duple x x hs ht h) = x
Module: Mathlib.Geometry.Convex.ConvexSpace.Defs

Goal state before:
pp: R : Type u_1
M : Type u_3
inst✝³ : PartialOrder R
inst✝² : Semiring R
inst✝¹ : IsStrictOrderedRing R
inst✝ : ConvexSpace R M
s t : R
hs : 0 ≤ s
ht : 0 ≤ t
h : s + t = 1
x y : M
⊢ convexCombPair s t hs ht h x y = convexCombPair t s ht hs ⋯ y x

Tactic:
unfold convexCombPair

Goal state after:
R : Type u_1
M : Type u_3
inst✝³ : PartialOrder R
inst✝² : Semiring R
inst✝¹ : IsStrictOrderedRing R
inst✝ : ConvexSpace R M
s t : R
hs : 0 ≤ s
ht : 0 ≤ t
h : s + t = 1
x y : M
⊢ sConvexComb (duple x y hs ht h) = sConvexComb (duple y x ht hs ⋯)
Module: Mathlib.Analysis.Convex.Quasiconvex

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
β : Type u_3
inst✝⁴ : Semiring 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : LinearOrder β
inst✝ : SMul 𝕜 E
s : Set E
f : E → β
hf : QuasiconvexOn 𝕜 s f
r : β
⊢ Convex 𝕜 {x | x ∈ s ∧ f x < r}

Tactic:
intro x

Goal state after:
𝕜 : Type u_1
E : Type u_2
β : Type u_3
inst✝⁴ : Semiring 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : LinearOrder β
inst✝ : SMul 𝕜 E
s : Set E
f : E → β
hf : QuasiconvexOn 𝕜 s f
r : β
x : E
⊢ x ∈ {x | x ∈ s ∧ f x < r} → StarConvex 𝕜 x {x | x ∈ s ∧ f x < r}